Supersymmetric Gauge Theories, Vortices and Equivariant Cohomology

We construct actions for (p, 0)-and (p, 1)-supersymmetric, 1 ≤ p ≤ 4, two-dimensional gauge theories coupled to non-linear sigma model matter with a Wess-Zumino term. We derive the scalar potential for a large class of these models. We then show that the Euclidean actions of the (2, 0) and (4, 0)-supersymmetric models without Wess-Zumino terms are bounded by topological charges which involve the equivariant extensions of the Kähler forms of the sigma model target spaces evaluated on the two-dimensional spacetime. We give similar bounds for Euclidean actions of appropriate gauge theories coupled to non-linear sigma model matter in higher spacetime dimensions which now involve the equivariant extensions of the Kähler forms of the sigma model target spaces and the second Chern character of gauge fields. The BPS configurations are generalisations of abelian and non-abelian vortices.